Powers and Exponent Laws PDF

2^UNIT^Powers and Exponent Laws What You'll Learn Use powers to show repeated multiplication. Evaluate powers with exponent 0. Write numbers using powers of 10. Use the order of operations with exponents. Use the exponent laws to simplify and evaluate expressions. Why It's Important...

2^UNIT^Powers and Exponent Laws What You'll Learn Use powers to show repeated multiplication. Evaluate powers with exponent 0. Write numbers using powers of 10. Use the order of operations with exponents. Use the exponent laws to simplify and evaluate expressions. Why It's Important Powers are used by lab technicians, when they interpret a patient's test results reporters, when they write large numbers in a news story Key Words integer opposite positive negative factor power base exponent squared cubed standard form product quotient 51 2.1 Skill Builder Multiplying Integers When multiplying 2 integers, look at the sign of each integer: When the integers have the same sign, their product is positive. When the integers have different signs, their product is negative. 6 ( 3) These 2 integers have different signs, so their product is negative. 6 ( 3) 18 **(** **) (** **)** **(** **)** ----- ( ) ----- ( ) **(** **)** ( ) ( ) ( 10) ( 2) These 2 integers have the same sign, so their product is positive. ( 10) ( 2) 20 *Check* 1\. Will the product be positive or negative? *When an integer is positive, we do not have to write the* *sign in* *front.* a\) 7 4 \_\_\_\_\_\_\_\_\_\_\_\_\_ b) 3 ( 6) \_\_\_\_\_\_\_\_\_\_\_\_\_ c) ( 9) 10 \_\_\_\_\_\_\_\_\_\_\_\_\_ d) ( 5) ( 9) \_\_\_\_\_\_\_\_\_\_\_\_\_ 2\. Multiply. a\) 7 4 \_\_\_\_\_\_\_\_ b) 3 (--6) \_\_\_\_\_\_\_\_ c) (--9) 10 \_\_\_\_\_\_\_\_ d) ( 5) ( 9) \_\_\_\_\_\_\_\_ e) (--3) (--5) \_\_\_\_\_\_\_\_ f) 2 ( 5) \_\_\_\_\_\_\_\_ g) (--8) 2 \_\_\_\_\_\_\_\_ h) ( 4) 3 \_\_\_\_\_\_\_\_ 52 Multiplying More than 2 Integers We can multiply more than 2 integers. Multiply pairs of integers, from left to right. ( 1) ( 2) (--3) ( 1) ( 2) ( 3) ( 4) 2 ( --3) 2 ( 3) ( 4) 6 ( 6) ( 4) 24 The product of 3 negative factors is The product of 4 negative factors is negative. positive. **Multiplying Integers** When the number of negative factors is *even*, the product is positive. When the number of negative factors is *odd*, the product is negative. We can show products of integers in different ways: ( 2) ( 2) 3 ( 2) is the same as ( 2)( 2)(3)( 2). So, ( 2) ( 2) 3 ( 2) ( 2)( 2)(3)( 2) 24 *Check* 1\. Multiply. a\) ( 3) ( 2) ( 1) 1 \_\_\_\_\_\_\_\_ *Is the answer* b\) ( 2)( 1)( 2)( 2)(2) \_\_\_\_\_\_\_\_ c\) ( 2)( 2)( 1)( 2)( 2) \_\_\_\_\_\_\_\_ d\) 3 3 2 \_\_\_\_\_\_\_\_ 53 *positive or* *negative? How can you tell?* 2.1 What Is a Power? FOCUS **Show repeated multiplication as a power.** We can use powers to show repeated multiplication. 2 2 2 2 2 2^5^ Repeated Power multiplication 5 **factors** of 2 We read 2^5^ as "2 to the 5th." Here are some other powers of 2. *2 is the base.* *5 is the exponent. 2^5^ is a power.* **Repeated Power Read as...** **Multiplication** 2 2^1^ 2 to the 1st 1 factor of 2 2 2 2^2^ 2 to the 2nd, or 2 squared 2 factors of 2 2 2 2 2^3^ 2 to the 3rd, or 2 cubed 3 factors of 2 2 2 2 2 2^4^ 2 to the 4th 4 factors of 2 *In each case, the exponent in the* *power is equal to the number of factors in the repeated* *multiplication.* ***Example 1*** Write as a power. **Writing Powers** a\) 4 4 4 4 4 4 b) 3 *Solution* a\) The base is 4. b) The base is 3. 4 4 4 4 4 4 4^6^ 3 6 factors of 4 1 factor of 3 So, 4 4 4 4 4 4 4^6^ So, 3 3^1^ 54 *Check* 1\. Write as a power. ~a)\ 2\ 2\ 2\ 2\ 2\ 2\ 2~\_\_\_\_ ~b)\ 5\ 5\ 5\ 5\ 5~\_\_\_\_ c\) ( 10)( 10)( 10) \_\_\_\_\_\_\_\_ d\) 4 4 \_\_\_\_\_\_ e\) ( 7)( 7)( 7)( 7)( 7)( 7)( 7)( 7) \_\_\_\_\_\_\_\_ 2\. Complete the table. **Repeated Multiplication Power Read as...** 8 8 8 8 \_\_\_\_\_\_\_\_\_ 8 to the 4th a) 7 7 \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b) 3 3 3 3 3 3 \_\_\_\_\_\_\_\_\_ 3 to the 6th c) 2 2 2 \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d) **Power Repeated Multiplication Standard Form** 2^5^ 2 2 2 2 2 32 ***Example 2* Evaluating Powers** Write as repeated multiplication and in standard form. a) 2^4^ b) 5^3^ *Solution* a\) 2^4^ 2 2 2 2 As repeated multiplication 16 Standard form b\) 5^3^ 5 5 5 As repeated multiplication 125 Standard form 55 *Check* 1\. Complete the table. **Power Repeated Multiplication Standard Form** 2^3^ 2 2 2 \_\_\_\_\_\_\_ 6^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 36 3^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ 10^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ ^8\ squared^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ ^7\ cubed^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ To evaluate a power that contains negative integers, identify the base of the power. Then, apply the rules for multiplying integers. ***Example 3* Evaluating Expressions Involving Negative Signs** Identify the base, then evaluate each power. a\) ( 5)^4^ b) 5^4^ *Solution* a\) ( 5)^4^ The brackets tell us that the base of this power is ( 5). ( 5)^4^ ( 5) ( 5) ( 5) ( 5) 625 There is an even number of negative integers, so the product is positive. b\) 5^4^ There are no brackets. So, the base of this power is 5. The negative sign applies to the whole expression. 5^4^ (5 5 5 5) 625 56 *Check* 1\. Identify the base of each power, then evaluate. a\) ( 1)^3^ b) 10^3^ The base is \_\_\_\_\_\_. The base is \_\_\_\_\_\_. ( 1)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_ 10^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) ( 7)^2^ d) ( 5)^4^ The base is \_\_\_\_\_\_. The base is \_\_\_\_\_\_. *The first negative sign applies to the whole expression.* ( 7)^2^ \_\_\_\_\_\_\_\_\_\_\_\_ ( 5)^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Practice 1\. Write as a power. a\) 8 8 8 8 8 8 8 7 factors of 8 The base is 8. There are \_\_\_\_\_\_ equal factors, so the exponent is \_\_\_\_\_\_. ~8\ 8\ 8\ 8\ 8\ 8\ 8\ 8~ \_\_\_\_ b\) 10 10 10 10 10 5 factors of 10 The base is \_\_\_\_\_\_. There are \_\_\_\_\_\_ equal factors, so the exponent is \_\_\_\_\_\_. So, 10 10 10 10 10 \_\_\_\_\_\_ c\) ( 2)( 2)( 2) 3 factors of \_\_\_\_\_\_ The base is \_\_\_\_\_\_. There are \_\_\_\_\_\_ equal factors, so the exponent is \_\_\_\_\_\_. So, ( 2)( 2)( 2) \_\_\_\_\_\_\_\_ d\) ( 13)( 13)( 13)( 13)( 13)( 13) \_\_\_\_\_\_\_\_ factors of \_\_\_\_\_\_\_\_ The base is \_\_\_\_\_\_\_\_. There are \_\_\_\_\_\_\_\_ equal factors, so the exponent is \_\_\_\_\_\_\_\_. So, ( 13)( 13)( 13)( 13)( 13)( 13) \_\_\_\_\_\_\_\_ 2\. Write each expression as a power. ~a)\ 9\ 9\ 9\ 9\ \_\_\_\_\_\_\ b)\ (5)(5)(5)(5)(5)(5)\ 5~\_\_\_ 4 c\) 11 11 \_\_\_\_\_\_ d) ( 12)( 12)( 12)( 12)( 12) \_\_\_\_\_\_\_\_ 57 3\. Write each power as repeated multiplication. a\) 3^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b) 3^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c) 2^7^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d\) 10^8^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Identify the base 4\. State whether the answer will be positive or negative. a) ( 3)^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_ b) 6^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_ c) ( 10)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_ d) 4^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_ 5\. Write each power as repeated multiplication and in standard form. a\) ( 3)^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_ b) 6^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_ c\) ( 10)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d) 4^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ 6\. Write each product as a power and in standard form. a\) ( 3)( 3)( 3) \_\_\_\_\_\_\_\_ b) ( 8)( 8) \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ first. *Predict.* *Will the answer be positive or* *negative?* c\) (8 8 8) \_\_\_\_\_\_\_\_ d) ( 1)( 1)( 1)( 1)( 1)( 1)( 1) \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ 7\. Identify any errors and correct them. a\) 4^3^ 12 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b\) (--2)^9^ is negative. \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) (--9)^2^ is negative. \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d\) 3^2^ 2^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ e\) ( 10)^2^ 100 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 58 2.2 Skill Builder Patterns and Relationships in Tables Look at the patterns in this table. **Input Output** 1 2 2 1 1 1 1 2 4 2 3 6 2 4 8 2 5 10 2 2 2 2 2 The input starts at 1 and increases by 1 each time. The output starts at 2 and increases by 2 each time. The input and output are also related. Double the input to get the output. *Check* 1\. a) Describe the patterns in the table. b\) What is the input in the last row? What is the output in the last row? **Input Output** 1 5 1 5 2 10 3 15 4 20 \_\_\_\_\_\_ \_\_\_\_\_\_ a\) The input starts at \_\_\_\_\_\_, and increases by \_\_\_\_\_\_ each time. The output starts at \_\_\_\_\_\_, and increases by \_\_\_\_\_\_ each time. You can also multiply the input by \_\_\_\_\_\_ to get the output. b\) The input in the last row is 4 \_\_\_\_\_\_ \_\_\_\_\_\_. The output in the last row is 20 \_\_\_\_\_\_ \_\_\_\_\_\_. 59 2\. a) Describe the patterns in the table. b\) Extend the table 3 more rows. **Input Output** 10 100 9 90 8 80 7 70 6 60 a\) The input starts at 10, and decreases by \_\_\_\_\_\_ each time. The output starts at 100, and decreases by \_\_\_\_\_\_ each time. You can also multiply the input by \_\_\_\_\_\_ to get the output. b\) To extend the table 3 more rows, continue to decrease the input by \_\_\_\_\_\_ each time. Decrease the output by \_\_\_\_\_\_ each time. **Input Output** 5 \_\_\_\_\_ \_\_\_\_\_ \_\_\_\_\_ \_\_\_\_\_ \_\_\_\_\_ Writing Numbers in Expanded Form 8000 is 8 thousands, or 8 1000 600 is 6 hundreds, or 6 100 50 is 5 tens, or 5 10 *Check* 1\. Write each number in expanded form. a) 7000 \_\_\_\_\_\_\_\_\_\_ b\) 900 \_\_\_\_\_\_\_\_\_\_ c\) 400 \_\_\_\_\_\_\_\_\_\_ d\) 30 \_\_\_\_\_\_\_\_\_\_ *Read it aloud.* 60 2.2 Powers of Ten and the Zero Exponent FOCUS **Explore patterns and powers of 10 to develop a meaning for the exponent 0.** This table shows decreasing powers of 3. **Power Repeated Multiplication Standard Form** 3^5^ 3 3 3 3 3 243 3 3^4^ 3 3 3 3 81 3 3^3^ 3 3 3 27 3 3^2^ 3 3 9 3 3^1\ 3^ 3 Look for patterns in the columns. The exponent decreases by 1 each time. Divide by 3 each time. The patterns suggest 3^0^ 1 because 3 3 1. We can make a similar table for the powers of any integer base except 0. **The Zero Exponent** A power with exponent 0 is equal to 1. ***Example 1* Powers with Exponent Zero** Evaluate each expression. *The base of the power can be any integer except 0.* a\) 6^0^ b) ( 5)^0^ *Solution* A power with exponent 0 is equal to 1. a\) 6^0^ 1 b) ( 5)^0^ 1 *Check* 1\. Evaluate each expression. a\) 8^0^ \_\_\_\_\_\_ b) 4^0^ \_\_\_\_\_\_ c) 4^0^ \_\_\_\_\_\_ d) ( 10)^0^ \_\_\_\_\_\_ *The zero exponent* *applies to the number in the brackets.* *If there are no brackets, the zero exponent applies only to the base.* 61 ***Example 2* Powers of Ten** Write as a power of 10. a\) 10 000 b) 1000 c) 100 d) 10 e) 1 *Solution* a\) 10 000 10 10 10 10 10^4^ b\) 1000 10 10 10 10^3^ c\) 100 10 10 10^2^ d\) 10 10^1^ e\) 1 10^0^ *Check* 1\. a) 5^1^ \_\_\_\_\_\_ b) ( 7)^1^ \_\_\_\_\_\_ c) 10^1^ \_\_\_\_\_\_ d) 10^0^ \_\_\_\_\_\_ Practice 1\. a) Complete the table below. **Power Repeated Multiplication Standard Form** 5^4^ 5 5 5 5 625 5^3^ 5 5 5 \_\_\_\_\_\_ 5^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ 5^1^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ b\) What is the value of 5^1^? \_\_\_\_\_\_ c\) Use the table. What is the value of 5^0^? \_\_\_\_\_\_ *Notice that the* *exponent is equal to the number of zeros.* 62 2\. Evaluate each power. a\) 2^0^ \_\_\_\_\_\_ b) 9^0^ \_\_\_\_\_\_ c) ( 2)^0^ \_\_\_\_\_\_ d) 2^0^ \_\_\_\_\_\_ e) 10^1^ \_\_\_\_\_\_ f) ( 8)^1^ \_\_\_\_\_\_ 3. Write each number as a power of 10. ~a)\ 10\ 000\ 10~^\_\_\_\_^ ~b)\ 1\ 000\ 000\ 10~\_\_\_\_ c) Ten million \_\_\_\_\_\_ d) One \_\_\_\_\_\_ e) 1 000 000 000 \_\_\_\_\_\_ f) 10 \_\_\_\_\_\_ 4. Evaluate each power of 10. a\) 10^6^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b) 10^0^ \_\_\_\_\_\_ c) 10^8^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d) 10^1^ \_\_\_\_\_\_ 5\. One trillion is written as 1 000 000 000 000. Write each number as a power of 10. a\) One trillion 1 000 000 000 000 \_\_\_\_\_\_\_\_ b\) Ten trillion 10 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ c) One hundred trillion \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ 6. Write each number in standard form. a\) 5 10^4^ 5 10 000 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *If there are no* *brackets, the* *exponent applies only to the base.* b\) (4 10^2^) (3 10^1^) (7 10^0^) (4 100) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) (2 10^3^) (6 10^2^) (4 10^1^) (9 10^0^) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d\) (7 10^3^) (8 10^0^) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 63 2.3 Skill Builder Adding Integers To add a positive integer and a negative integer: 7 ( 4) Model each integer with tiles. Circle zero pairs. 7: --4: There are 4 zero pairs. There are 3 tiles left. They model 3. So, 7 ( 4) 3 To add 2 negative integers: ( 4) ( 2) Model each integer with tiles. Combine the tiles. --4: --2: There are 6 tiles. They model 6. So, ( 4) ( 2) --6 *Check* 1\. Add. a\) ( 3) ( 4) \_\_\_\_\_\_ b) 6 ( 2) \_\_\_\_\_\_ *Each pair of 1 tile and 1 tile makes a zero pair. The pair models 0.* c\) ( 5) 2 \_\_\_\_\_\_ d) ( 4) ( 4) \_\_\_\_\_\_ 2\. a) Kerry borrows \$5. Then she borrows another \$5. Add to show what Kerry owes. ( 5) ( 5) \_\_\_\_\_\_ Kerry owes \$\_\_\_\_\_\_. b\) The temperature was 8°C. It fell 10°C. Add to show the new temperature. *When an amount of money is* *negative, it is owed.* 8 \_\_\_\_\_\_ \_\_\_\_\_\_ The new temperature is \_\_\_\_\_\_°C. ( ) 64 Subtracting Integers To subtract 2 integers: 3 6 Model the first integer. Take away the number of tiles equal to the second integer. Model 3. There are not enough tiles to take away 6. To take away 6, we need 3 more tiles. We add zero pairs. Add 3 tiles and 3 tiles. Now take away the 6 tiles. Since 3 tiles remain, we write: 3 6 3 *Adding zero pairs does not change the value. Zero pairs represent 0.* When tiles are not available, think of subtraction as the opposite of addition. To subtract an integer, add its opposite integer. For example, ( 3) ( 2) 5 ( 3) ( 2) 5 Subtract 2. Add --2. *Check* 1\. Subtract. a\) ( 6) 2 \_\_\_\_\_\_ b) 2 ( 6) \_\_\_\_\_\_ c) ( 8) 9 \_\_\_\_\_\_ d) 8 ( 9) \_\_\_\_\_\_ 65 Dividing Integers When dividing 2 integers, look at the sign of each integer: When the integers have the same sign, their quotient is positive. When the integers have different signs, their quotient is negative. *The same rule applies to the multiplication of integers.* 6 ( 3) These 2 integers have different signs, so their quotient is negative. 6 ( 3) 2 ( 10) ( 2) These 2 integers have the same sign, so their quotient is positive. ( 10) ( 2) 5 *Check* 1\. Calculate. a\) ( 4) 2 \_\_\_\_\_\_ b\) ( 6) ( 3) \_\_\_\_\_\_ c\) 15 ( 3) \_\_\_\_\_\_ 66 2.3 Order of Operations with Powers FOCUS **Explain and apply the order of operations with exponents.** We use this order of operations when evaluating an expression with powers: Do the operations in brackets first. Evaluate the powers. Multiply and divide, in order, from left to right. Add and subtract, in order, from left to right. We can use the word BEDMAS to help us remember the order of operations: **B B**rackets **E E**xponents **D D**ivision **M M**ultiplication **A A**ddition **S S**ubtraction ***Example 1* Adding and Subtracting with Powers** Evaluate. a\) 2^3^ 1 b) 8 3^2^ c) (3 1)^3^ *Solution* a\) 2^3^ + 1 Evaluate the power first: 2^3^ (2)(2)(2) 1 Multiply: (2)(2)(2) 8 1 Then add: 8 1 9 b\) 8 3^2^ Evaluate the power first: 3^2^ 8 (3)(3) Multiply: (3)(3) 8 9 Then subtract: 8 9 1 c\) (3 1)^3^ Subtract inside the brackets first: 3 1 2^3^ Evaluate the power: 2^3^ (2)(2)(2) Multiply: (2)(2)(2) 8 *To subtract, add the* *opposite: 8* *(* *9)* 67 *Check* 1\. Evaluate. a\) 4^2^ 3 \_\_\_\_\_\_\_ 3 b) 5^2^ 2^2^ \_\_\_\_\_\_ (2)(2) \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ c\) (2 1)^2^ \_\_\_\_\_\_\_\_^2^ d) (5 6)^2^ \_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_ ***Example 2* Multiplying and Dividing with Powers** Evaluate. a\) \[2 ( 2)^3^\]^2^ b) (7^2^ 5^0^) ( 5)^1^ Curved brackets Square brackets *When we need curved* *brackets for integers, we use* *square brackets to show the* *order of operations.* *Solution* a\) \[2 ( 2)^3^\]^2^ Evaluate what is inside the square brackets first: 2 ( 2)^3^ \[2 ( 8)\]^2^ Start with ( 2)^3^ 8. ( 16)^2^ 256 b\) (7^2^ 5^0^) ( 5)^1^ Evaluate what is inside the brackets first: 7^2^ 5^0^ (49 1) ( 5)^1^ Add inside the brackets: 49 1 50 ( 5)^1^ Evaluate the power: ( 5)^1^ 50 ( 5) 10 68 *Check* 1\. Evaluate. a\) 5 3^2^ 5 \_\_\_\_\_\_ \_\_\_\_\_\_ b) 8^2^ 4 \_\_\_\_\_\_ \_\_\_\_\_\_ 4 5 \_\_\_\_\_\_ \_\_\_\_\_\_ 4 \_\_\_\_\_\_ \_\_\_\_\_\_ c\) (3^2^ 6^0^)^2^ 2^1^ d) 10^2^ (2 2^2^)^2^ 10^2^ (2 \_\_\_\_\_)^2^ (\_\_\_\_\_\_ \_\_\_\_\_\_)^2^ 2^1^ 10^2^ \_\_\_\_\_\_ \_\_\_\_\_\_ 2^1^ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ ***Example 3* Solving Problems Using Powers** Corin answered the following skill-testing question to win free movie tickets: 120 20^3^ 10^3^ 12 120 His answer was 1568. Did Corin win the movie tickets? Show your work. *Solution* 120 20^3^ 10^3^ 12 120 Evaluate the powers first: 20^3^ and 10^3^ 120 8000 1000 12 120 Divide and multiply. 120 8 1440 Add: 120 8 1440 1568 Corin won the movie tickets. *Check* 1\. Answer the following skill-testing question to enter a draw for a Caribbean cruise. (6 4) 3^2^ 10 10^2^ 4 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 69 Practice 1\. Evaluate. a\) 2^2^ 1 \_\_\_\_\_\_\_ 1 b) 2^2^ 1 \_\_\_\_\_\_\_ 1 \_\_\_\_\_\_ 1 \_\_\_\_\_\_ 1 \_\_\_\_\_\_ \_\_\_\_\_\_ c\) (2 1)^2^ \_\_\_\_\_\_ d) (2 1)^2^ \_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ 2\. Evaluate. a\) 4 2^2^ 4 \_\_\_\_\_\_\_ b) 4^2^ 2 \_\_\_\_\_\_\_ 2 4 \_\_\_\_\_ \_\_\_\_\_\_ 2 \_\_\_\_\_ \_\_\_\_\_\_ c\) (4 2)^2^ \_\_\_\_\_\_ d) ( 4)^2^ 2 \_\_\_\_\_\_\_\_\_\_\_ 2 \_\_\_\_\_\_\_ \_\_\_\_\_\_ 2 \_\_\_\_\_\_ \_\_\_\_\_\_ 3\. Evaluate. a\) 2^3^ ( 1)^3^ \_\_\_\_\_\_\_\_\_ ( 1)^3^ b) (2 1)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ ( 1)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ c\) 2^3^ ( 1)^3^ \_\_\_\_\_\_\_\_\_ ( 1)^3^ d) (2 1)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ ( 1)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ 4\. Evaluate. a\) 3^2^ (--1)^2^ \_\_\_\_\_\_\_ (--1)^2^ b) (3 1)^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_^2^ \_\_\_\_\_\_\_ (--1)^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_ c\) 3^2^ ( 2)^2^ \_\_\_\_\_\_\_ ( 2)^2^ d) 5^2^ ( 5)^1^ \_\_\_\_\_\_\_ ( 5)^1^ \_\_\_\_\_\_\_ ( 2)^2^ \_\_\_\_\_\_\_ ( 5)^1^ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 70 5\. Evaluate. a\) ( 2)^0^ ( 2) \_\_\_\_\_\_ ( 2) b) 2^3^ ( 2)^2^ \_\_\_\_\_\_\_\_\_ ( 2)^2^ \_\_\_\_\_\_ \_\_\_\_\_\_ ( 2)^2^ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ c\) (3 2)^0^ (3 2)^0^ \_\_\_\_\_\_ \_\_\_\_\_\_ d) (3 5^2^)^0^ \_\_\_\_\_\_ \_\_\_\_\_\_ e\) (2)(3) -- (4)^2^ (2)(3) -- \_\_\_\_\_\_\_ f) 3(2 1)^2^ 3\_\_\_\_\_\_ (2)(3) -- \_\_\_\_\_\_ 3\_\_\_\_\_\_ \_\_\_\_\_\_ -- \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ *A power with exponent 0 is equal to 1.* g\) ( 2)^2^ (3)(4) \_\_\_\_\_\_\_\_\_\_ (3)(4) h) ( 2) 3^0^ ( 2) ( 2) \_\_\_\_\_\_ ( 2) \_\_\_\_\_\_ (3)(4) ( 2) \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ 6\. Amaya wants to replace the hardwood floor in her house. Here is how she calculates the cost, in dollars: 70 6^2^ 60 6^2^ How much will it cost Amaya to replace the hardwood floor? 70 \_\_\_\_\_\_\_ 60 \_\_\_\_\_\_\_ 70 \_\_\_\_\_\_ 60 \_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_ It will cost Amaya \$\_\_\_\_\_\_\_ to replace the hardwood floor. *Remember the order of operations: BEDMAS* 71 CHE[CKP~~O~~]INT CHE[CKP~~O~~]INT *Can you...* Use powers to show repeated multiplication? Use patterns to evaluate a power with exponent zero, such as 50? Use the correct order of operations with powers? 1\. Give the base and exponent of each power. 2.1 a\) 6^2^ Base: \_\_\_\_\_\_ Exponent: \_\_\_\_\_\_ There are \_\_\_\_\_\_ factors of \_\_\_\_\_\_. b\) 4^5^ Base: \_\_\_\_\_\_ Exponent: \_\_\_\_\_\_ There are \_\_\_\_\_\_ factors of \_\_\_\_\_\_. c\) ( 3)^8^ Base: \_\_\_\_\_\_ Exponent: \_\_\_\_\_\_ There are \_\_\_\_\_\_ factors of \_\_\_\_\_\_. d\) 3^8^ Base: \_\_\_\_\_\_ Exponent: \_\_\_\_\_\_ There are \_\_\_\_\_\_ factors of \_\_\_\_\_\_. 2\. Write as a power. ~a)\ 7\ 7\ 7\ 7\ 7\ 7\ 7~\_\_\_\_ ~b)\ 2\ 2\ 2\ 2\ 2~\_\_\_\_ c\) 5 \_\_\_\_\_\_ d\) ( 5)( 5)( 5)( 5)( 5) \_\_\_\_\_\_ 3\. Write each power as repeated multiplication and in standard form. a) 5^2^ 5 \_\_\_\_\_\_ \_\_\_\_\_\_ b\) 2^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ c\) 3^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ 72 4\. a) 2.2 Complete the table. **Power Repeated Multiplication Standard Form** 7^3^ 7 7 7 343 7^2^ 7 7 7^1^ b\) What is the value of 7^0^? \_\_\_\_\_\_ 5\. Write each number in standard form and as a power of 10. a\) One hundred 100 b) Ten thousand **\_\_\_\_\_\_\_\_\_\_\_\_\_** ~10~^\_\_\_^ ~10~\_\_\_\_ c\) One million **\_\_\_\_\_\_\_\_\_\_\_\_\_** d) One \_\_\_ ~10~^\_\_\_^ ~10~\_\_\_\_ 6\. Evaluate. a\) 6^0^ \_\_\_\_\_\_ b) ( 8)^0^ \_\_\_\_\_\_ c\) 12^1^ \_\_\_\_\_\_ d) 8^0^ \_\_\_\_\_\_ 7\. Write each number in standard form. a\) 4 10^3^ 4 \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_ b\) (1 10^3^) (3 10^2^) (2 10^1^) (1 10^0^) (1 1000) (3 \_\_\_\_\_\_) (**\_\_\_\_\_\_\_\_\_**) ( **\_\_\_\_\_\_\_\_\_**) **\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ **\_\_\_\_\_\_\_\_\_** c\) (4 10^3^) (2 10^2^) (3 10^1^) (6 10^0^) (4 **\_\_\_\_\_\_\_\_\_**) (**\_\_\_\_\_\_\_\_\_**) (**\_\_\_\_\_\_\_\_\_**) (**\_\_\_\_\_\_\_\_\_**) **\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ **\_\_\_\_\_\_\_\_\_** d\) (8 10^2^) (1 10^1^) (9 10^0^) **\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** 73 8\. Evaluate. 2.3 a\) 3^2^ 5 \_\_\_\_\_\_\_ 5 b) 5^2^ 2^3^ \_\_\_\_\_\_\_ 2^3^ \_\_\_\_\_\_ 5 \_\_\_\_\_\_ 2^3^ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_ c\) (2 3)^3^ (\_\_)^3^ d) 2^3^ ( 3)^3^ **~\_\_\_\_\_\_\_\_~** ( 3)^3^ **\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ ( 3)^3^ **\_\_\_\_\_** \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ 9\. Evaluate. a\) 5 3^2^ 5 **\_\_\_\_\_** b) 8^2^ 4 **\_\_\_\_\_** 4 **\_\_\_\_\_** **\_\_\_\_\_** c\) (10 2) 2^2^ **\_\_\_\_\_** 2^2^ d) (7^2^ 1) (2^3^ 2) **\_\_\_\_\_** **\_\_\_\_\_** (**\_\_\_\_\_** 1) (**\_\_\_\_\_** 2) **\_\_\_\_\_** **\_\_\_\_\_** **\_\_\_\_\_** **\_\_\_\_\_** 10\. Evaluate. State which operation you do first. a\) 3^2^ 4^2^ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_ b\) \[( 3) 2\]^3^ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** (\_\_\_\_\_\_)^3^ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ c\) ( 2)^3^ ( 3)^0^ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ d\) \[(6 3)^3^ (2 2)^2^\]^0^ **\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** \_\_\_\_\_\_ 74 2.4 Skill Builder Simplifying Fractions To simplify a fraction, divide the numerator and denominator by their common factors. 5 5 5 5 To simplify : 5 5 *This fraction shows repeated multiplication.* Divide the numerator and denominator by their common factors: 5 5. 1 1 [5 5 5 5 ] ~~1~~ 1 5 5 [5 5] 1 25 *Check* 1\. Simplify each fraction. 3 3 3 a) 3 \_\_\_\_\_\_ \_\_\_\_\_ *What are the common* *factors?* b\) c) d) 8 8 8 8 8 8 8 8 8 8 \_\_\_\_\_ [5 5 5 5 5 ] 5 5 5 \_\_\_\_\_\_ \_\_\_\_\_ [2 2 2 2 2 2 2 2 ] 2 2 2 2 2 \_\_\_\_\_\_\_\_\_ \_\_\_\_\_ 75 2.4 Exponent Laws I FOCUS **Understand and apply the exponent laws for products and quotients of powers.** Multiply 3^2^ 3^4^. 3^2^ 3^4^ Write as repeated multiplication. (3 3) (3 3 3 3) 2 factors of 3 4 factors of 3 3 3 3 3 3 3 6 factors of 3 3^6^ Base Exponent So, 3**^2^** 3**^4^** 3**^6^** Look at the pattern in the exponents. **2** **4** **6** We write: 3**^2^** 3**^4^** 3^**(2**\ **4)**^ 3**^6^** This relationship is true when you multiply any 2 powers with the same base. **Exponent Law for a Product of Powers** To multiply powers with the same base, add the exponents. ***Example 1* Simplifying Products with the Same Base** Write as a power. a\) 5^3^ 5^4^ b) ( 6)^2^ ( 6)^3^ c) (7^2^)(7) *Solution* a\) The powers have the same base: 5 Use the exponent law for products: add the exponents. 5^3^ 5^4^ 5^(3\ 4)^ 5^7^ *To check your work, you can write the powers as repeated multiplication.* 76 b\) The powers have the same base: 6 ( 6)^2^ ( 6)^3^ ( 6)^(2\ 3)^ Add the exponents. ( 6)^5^ c\) (7^2^)(7) 7^2^ 7^1^ Use the exponent law for products. 7^(2\ 1)^ Add the exponents. 7^3^ *Check* 1\. Write as a power. a\) 2^5^ 2^4^ 2^(\ )^ b) 5^2^ 5^5^ 5 2 5 c\) ( 3)^2^ ( 3)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_ d) 10^5^ 10 \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ Divide 3^4^ 3^2^. *7 can be written as 7^1^.* ^4^ 3^4^ 3^2^ ~~.~~ 3^2^ [3 3 3 3 ] Simplify. 3 3 1 1 *We can show division in fraction form.* [3 3 3 3] ~~1~~ 1 3 3 [3 3 ] 1 3 3 3^2^ So, 3**^4^** 3**^2^** 3**^2^** Look at the pattern in the exponents. **4** **2** **2** We write: 3**^4^** 3**^2^** 3^(**4**\ --\ **2**)^ 3**^2^** This relationship is true when you divide any 2 powers with the same base. 77 **Exponent Law for a Quotient of Powers** To divide powers with the same base, subtract the exponents. ***Example 2* Simplifying Quotients with the Same Base** Write as a power. a\) 4^5^ 4^3^ b) ( 2)^7^ ( 2)^2^ *Solution* Use the exponent law for quotients: subtract the exponents. a\) 4^5^ 4^3^ 4^(5\ 3)^ 4^2^ b\) ( 2)^7^ ( 2)^2^ ( 2)^(7\ 2)^ ( 2)^5^ *Check* 1\. Write as a power. a\) ( 5)^6^ ( 5)^3^ ( 5)^\_\_\_\_\_\_\_^ \_\_\_\_\_\_\_ [( 3)]^9^ b\) ( 3)^\_\_\_\_\_\_\_\_^ ( 3)^5^ The powers have the same base: 4 To check your work, you can write the powers as repeated multiplication. The powers have the same base: 2 *(* *3)^9^* \_\_\_\_\_\_\_ c\) 8^4^ 8^3^ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_ d\) 9^8^ 9^2^ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_ *(* *3)^5^* *is the same as (* *3)^9^* *(* *3)^5^* 78 ***Example 3* Evaluating Expressions Using Exponent Laws** Evaluate. a\) 2^2^ 2^3^ 2^4^ b) ( 2)^5^ ( 2)^3^ ( 2) *Solution* a\) 2^2^ 2^3^ 2^4^ Add the exponents of the 2 powers that are multiplied. 2^(2\ 3)^ 2^4^ Then, subtract the exponent of the power that is divided. 2^5^ 2^4^ 2^(5\ 4)^ 2^1^ 2 b\) ( 2)^5^ ( 2)^3^ ( 2) Subtract the exponents of the 2 powers that are divided. ( 2)^(5\ 3)^ ( 2) ( 2)^2^ ( 2) Multiply: add the exponents. ( 2)^(2\ 1)^ ( 2)^(3)^ ( 2)( 2)( 2) 8 *Check* 1\. Evaluate. a\) 4 4^3^ 4^2^ ~4~ ^\_\_\_\_\ \_\_\_\_^ 4^2^ b) ( 3) ( 3) ( 3) ( ) ~4~^\_\_\_\_^ 4^2^ ( 3) ^\_\_\_\_\_\_\_\_^ ( 3) ~4~ ^\_\_\_\_\ \_\_\_\_^ ~(\ 3)~^\_\_\_^ ( 3) ( ) ~4~^\_\_\_\_^ ( 3)^\_\_\_\_\_\_\_\_^ \_\_\_\_\_\_ ~(\ 3)~\_\_\_\_ \_\_\_\_\_\_ 79 *(* *3)* *(* *3)^1^* Practice 1\. Write each product as a single power. a\) 7^6^ 7^2^ 7^(\_\_\_\_\ \_\_\_)^ b) ( 4)^5^ ( 4)^3^ ( 4)^\_\_\_\_\_\_\_\_^ ~7~^\_\_\_^ ~(\ 4)~\_\_\_\_ c\) (--2) ( 2)^3^ \_\_\_\_\_\_\_\_\_\_\_\_ d) 10^5^ 10^5^ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ e\) 7^0^ 7^1^ \_\_\_\_\_\_\_\_\_\_\_\_ f) ( 3)^4^ ( 3)^5^ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ 2\. Write each quotient as a power. a\) ( 3)^5^ ( 3)^2^ ( 3)^(\_\_\_\ \_\_\_\_)^ b) 5^6^ 5^4^ 5^\_\_\_\_\_\_\_\_^ *To multiply* *powers with the same base, add the exponents.* ~(\ 3)~^\_\_\_^ ~5~\_\_\_\_ ^7^ ^8^ *To divide powers with the same* ~c)\ 4~^\_\_\_\_\_\_^ d) \_\_\_\_\_\_\_\_ *base, subtract* 4^4^ 5^6^ ~4~^\_\_\_\_^ \_\_\_\_\_\_ [( 6)]^8^ *the exponents.* e\) 6^4^ 6^4^ \_\_\_\_\_\_\_ f) \_\_\_\_\_\_\_\_\_\_\_\_ ( 6)^7^ \_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ 3. Write as a single power. ^2^ ^2^ ~~ a\) 2^3^ 2^4^ 2^5^ 2^(\_\_\_\_\ \_\_\_\_)^ 2^5^ b) ~2~^\_\_\_^ 2^5^ 3^2^ 3^2^ 3 ~~ *Which exponent law should you* 2^\_\_\_\_\_\_\_\_^ ~3~ *use?* ^\_\_\_\_\_\_\_\_\_\_\_\_^ ~2~^\_\_\_\_\_\_\_^ \_\_\_\_\_\_\_\_\_\_\_\_ c\) 10^3^ 10^5^ 10^2^ \_\_\_\_\_\_\_\_\_ 10^2^ d) ( 1)^9^ ( 1)^5^ ( 1)^0^ \_\_\_\_\_\_ 10^2^ \_\_\_\_\_\_\_\_\_\_\_\_ (--1)^0^ \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ ( 1)^0^ \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ 80 4\. Simplify, then evaluate. a\) ( 3)^1^ ( 3)^2^ 2 b) 9^9^ 9^7^ 9^0^ \_\_\_\_\_\_\_\_\_\_\_\_ 9^0^ \_\_\_\_\_\_\_\_\_\_\_\_ 2 \_\_\_\_\_\_ 9^0^ \_\_\_\_\_\_\_ 2 \_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_ 2 \_\_\_\_\_\_ \_\_\_\_\_\_\_ \_\_\_\_\_\_ *See if you can use the exponent laws to simplify.* ^2^ ^5^ c\) \_\_\_\_\_\_\_\_\_\_\_\_ d) 5 5^\_\_\_\_\_\_\_\_^ 5 5^0^ 5^4^ \_\_\_\_\_\_\_\_\_\_\_\_ ~5~^\_\_\_\_^ 5 \_\_\_\_\_\_\_\_\_\_\_\_ 5^\_\_\_\_\_\_\_\_^ ~5~\_\_\_\_ \_\_\_\_\_\_ 5\. Identify any errors and correct them. a\) 4^3^ 4^5^ 4^8^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b\) 2^5^ 2^5^ 2^25^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) ( 3)^6^ ( 3)^2^ ( 3)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d\) 7^0^ 7^2^ 7^0^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ e\) 6^2^ 6^2^ 6^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ f\) 10^6^ 10 10^6^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ g\) 2^3^ 5^2^ 10^5^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 81 2.5 Skill Builder Grouping Equal Factors In multiplication, you can group equal factors. For example: 3 7 7 3 7 7 3 Group equal factors. *Order does not matter in* *multiplication.* 3 3 3 7 7 7 7 Write repeated multiplication as powers. 3^3^ 7^4^ *Check* 1\. Group equal factors and write as powers. 2 2 2 a\) 2 10 2 10 2 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b\) 2 5 2 5 2 5 2 5 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Multiplying Fractions To multiply fractions, first multiply the numerators, and then multiply the denominators. [2 ] [2 ] [2 ] [2 ] [2 2 2 2 ] Write repeated multiplication as powers. 3 3 *Check* 3 3 3 3 3 3 ^4^ 3^4^ *There are 4 factors of 2, and 4 factors of 3.* 1\. Multiply the fractions. Write as powers. [3 ] [3 ] [3 ] [1 ] [1 ] [1 ] [1 ] [1 ] [1 ] a\) b) 4 4 4 2 2 2 2 2 2 82 2.5 Exponent Laws II FOCUS **Understand and apply exponent laws for powers of: products; quotients; and powers.** Multiply 3^2^ 3^2^ 3^2^. Use the exponent law for the product of powers. 3^2^ 3^2^ 3^2^ 3^2\ 2\ 2^ Add the exponents. 3^6^ We can write repeated multiplication as powers. So, 3^2^ 3^2^ 3^2^ 3 factors of (3^2^) The base is 3^2^. The exponent is 3. (3**^2^**)**^3^** This is a **power of a power.** *This is also a power.* 3**^6^** Look at the pattern in the exponents. **2** **3** **6** We write: (3**^2^**)**^3^** 3^**2**\ **3**^ 3**^6^** **Exponent Law for a Power of a Power** To raise a power to a power, multiply the exponents. For example: (2^3^)^5^ 2^3\ 5^ ***Example 1* Simplifying a Power of a Power** Write as a power. a\) (3^2^)^4^ b) \[( 5)^3^\]^2^ c) (2^3^)^4^ *Solution* Use the exponent law for a power of a power: multiply the exponents. a\) (3^2^)^4^ 3^2\ 4^ 3^8^ b\) \[( 5)^3^\]^2^ ( 5)^3\ 2^ The base is 5. ( 5)^6^ c\) (2^3^)^4^ (2^3\ 4^) The base is 2. 2^12^ 83 *Check* 1\. Write as a power. a\) (9^3^)^4^ ~9~^\_\_\_\_\ \_\_\_\_^ b) \[( 2)^5^\]^3^ ( 2)^\_\_\_\_\_\_\_^ c) (5^4^)^2^ (5^\_\_\_\_\_\_\_^) ~9~^\_\_\_\_^ ~(\ 2)~^\_\_\_^ ~5~\_\_\_\_ Multiply (3 4)^2^. The base of the power is a product: 3 4 Write as repeated multiplication. (3 4)^2^ (3 4) (3 4) Remove the brackets. 3 4 3 4 Group equal factors. **base** (3 3) (4 4) Write as powers. 2 factors of 3 2 factors of 4 3^2^ 4^2^ So, (3 4)^2^ 3^2^ 4^2^ **power power product** **Exponent Law for a Power of a Product** The power of a product is the product of powers. For example: (2 3)^4^ 2^4^ 3^4^ ***Example 2* Evaluating Powers of Products** Evaluate. a\) (2 5)^2^ b) \[( 3) 4\]^2^ *Solution* Use the exponent law for a power of a product. a\) (2 5)^2^ 2^2^ 5^2^ b) \[( 3) 4\]^2^ ( 3)^2^ 4^2^ (2)(2) (5)(5) ( 3)( 3) (4)(4) 4 25 9 16 100 144 Or, use the order of operations and evaluate what is inside the brackets first. a\) (2 5)^2^ 10^2^ b) \[( 3) 4\]^2^ ( 12)^2^ 100 144 84 *Check* **1.** Write as a product of powers. a\) (5 7)^4^ \_\_\_\_\_\_ \_\_\_\_\_\_ b) (8 2)^2^ \_\_\_\_\_\_ \_\_\_\_\_\_ **2.** Evaluate. a\) \[( 1) 6\]^2^ \_\_\_\_\_\_\_^2^ b) \[( 1) ( 4)\]^3^ \_\_\_\_\_\_^3^ \_\_\_\_\_\_\_ \_\_\_\_\_\_ ~4\ a~ ^^ ~4~~~b~~^2^ [3 ] Evaluate. The base of the power is a quotient: **base** Write as repeated multiplication. ~a~^^~4\ a\ ~~b~~~ ^^~4\ a\ ~~b~~~ ^^~4~~~b~~^2^ [3 ] [3 ] Multiply the fractions. 4 4 [3 3 ] Write repeated multiplication as powers. 4 4 ^2^ 4^2^ **power** 4^2^ ~a~^^~4~~~b~~^2^ So, ^2^ **quotient power** **Exponent Law for a Power of a Quotient** The power of a quotient is the quotient of powers. 3^4^ ~a~^^~3~~~b~~^4^ For example: ^4^ ***Example 3* Evaluating Powers of Quotients** Evaluate. a\) \[30 ( 5)\]^2^ ~b)\ a~^^~4~ ~~b~~^2^ 85 *Solution* Use the exponent law for a power of a quotient. ~a~ ^^ ~5~~~b~~^2^ 4^2^ ~a~^^~4~ ~~b~~^2^ a\) \[30 ( 5)\]^2^ b) ^2^ ( 5)^2^ [900 ] ^2^ 16 25 25 36 Or, use the order of operations and evaluate what is inside the brackets first. ~a~^^~4~ ~~b~~^2^ a\) \[30 ( 5)\]^2^ ( 6)^2^ b) 5^2^ 36 25 *Check* 1\. Write as a quotient of powers. ~a~^^~4~~~b~~^5^ ^a)^ ~\_\_\_\_\_\_\_\_~ b) \[1 ( 10)\]^3^ \_\_\_\_\_\_\_\_ 2. Evaluate. ~a~^^~6~ ~~b~~^3^ a\) \[( 16) ( 4)\]^2^ b) \_\_\_\_\_\_ \_\_\_\_\_\_^2^ \_\_\_\_\_\_ \_\_\_\_\_\_ Practice 1\. Write as a product of powers. a\) (5 2)^4^ ~5~^\_\_\_\_^ ~2~^\_\_\_\_^ b) (12 13)^2^ \_\_\_\_\_\_\_\_\_\_\_\_ *You can* *evaluate what is inside the brackets first.* c\) \[3 ( 2)\]^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_ d) \[( 4) ( 5)\]^5^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2\. Write as a quotient of powers. a\) (5 8)^0^ ~\_\_\_\_\_\_\_~ b) \[( 6) 5\]^7^ \_\_\_\_\_\_\_ ~a~^^~5~~~b~~^2^ 3 ^c)^ ~\_\_\_\_\_\_\_~ ^d)^ ~\_\_\_\_\_\_\_\ a~ ~2~~~b~~ 86 3\. Write as a power. a\) (5^2^)^3^ ~5~^\_\_\_\_\ \_\_\_\_^ b) \[( 2)^3^\]^5^ ( 2)^\_\_\_\_\_\_^ ~5~^\_\_\_\_^ \_\_\_\_\_\_\_\_ c\) (4^4^)^1^ \_\_\_\_\_\_\_\_ d) (8^0^)^3^ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ 4\. Evaluate. **[^\_\_\_^ ]** a\) \[(6 ( 2)\]^2^ \_\_\_\_\_\_\_\_\_\_\_\_ b) (3 4)^2^ (\_\_\_\_\_) \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_ ~a~ [8 ] 2 c\) \_\_\_\_\_\_\_\_\_\_\_\_ d) (10 3)^1^ \_\_\_\_\_\_\_\_\_\_\_\_ ~2~~~b~~ ~\_\_\_\_\_\_\_\_\_\_\_\_~ \_\_\_\_\_\_\_\_\_\_\_\_ e\) \[( 2)^1^\]^2^ \_\_\_\_\_\_\_\_\_\_\_\_ f) \[( 2)^1^\]^3^ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_ 5\. Find any errors and correct them. a\) (3^2^)^3^ 3^5^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b\) (3 2)^2^ 3^2^ 2^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) (5^3^)^3^ 5^9^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ^8^ [2 ] ^8^ ^d)^ ( ~~)~~ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 3 3^8^ e\) (3 2)^2^ 36 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ^2^ [2 ] ^f)^ ( ~~)~~ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ [4 ] 3 6 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ g\) \[( 3)^3^\]^0^ ( 3)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ h\) \[( 2) ( 3)\]^4^ 6^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 87 Unit 2 Puzzle **Bird's Eye View** This is a view through the eyes of a bird. What does the bird see? To find out, simplify or evaluate each expression on the left, then find the answer on the right. Write the corresponding letter beside the question number. The numbers at the bottom of the page are question numbers. Write the corresponding letter over each number. 1\. 5 5 5 5 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ A 100 000 2\. 2^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ P 5^6^ ^6^ 3\. \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ S 0 3^2^ 4\. 4 4 4 4 4 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ E 1 5\. ( 2)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ F 3^4^ 6\. ( 2) 4 2 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ G 6 7\. (5^2^)^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ I 8 8\. 3^2^ 2^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ O 4^6^ 9\. 10^2^ 10^3^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ N 4^5^ 10\. 5 3^0^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ R 5^4^ 11\. 4^7^ 4 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Y 8 \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_ \_\_\_\_ \_\_\_\_ \_\_\_\_ \_\_\_\_ \_\_\_ \_\_\_ 9 7 8 1 6 11 4 3 1 5 2 4 10 9 4 8 10 10 88 Unit 2 Study Guide **Skill Description Example** Evaluate a power with an integer base. Evaluate a power with an exponent 0. Use the order of operations to evaluate expressions containing exponents. Apply the exponent law for a product of powers. Write the power as repeated multiplication, then evaluate. A power with an integer base and an exponent 0 is equal to 1. Evaluate what is inside the brackets. Evaluate powers. Multiply and divide, in order, from left to right. Add and subtract, in order, from left to right. To multiply powers with the same base, add the exponents. ( 2)^3^ ( 2) ( 2) ( 2) 8 8^0^ 1 (3^2^ 2) ( 5) (9 2) ( 5) \(11) ( 5) 55 4^3^ 4^6^ 4^3\ 6^ 4^9^ Apply the exponent law for a quotient of powers. To divide powers with the same base, subtract the 2^7^ 2^4^ ^7^ 2^4^ 2^7\ 4^ 2^3^ Apply the exponent law for exponents. To raise a power to a power, a power of a power. ~multiply\ the\ exponents.~(5^3^)^2^ 5^3\ 2^ 5^6^ Apply the exponent law for a power of a product. Apply the exponent law for a power of a quotient. Write the power of a product as a product of powers. Write the power of a quotient as a quotient of powers. (6 3)^5^ 6^5^ 3^5^ ~a~^^ ~4~~~b~~^2^ ^2^ 4^2^ 89 Unit 2 Review 1\. Give the base and exponent of each power. 2.1 a\) 6^2^ Base **\_\_\_\_\_\_** Exponent **\_\_\_\_\_\_** b\) ( 3)^8^ Base **\_\_\_\_\_\_** Exponent **\_\_\_\_\_\_** 2\. Write as a power. ~a)\ 4\ 4\ 4\ 4~^\_\_\_\_^ b) ( 3)( 3)( 3)( 3)( 3) **\_\_\_\_\_\_\_\_** 3. Write each power as repeated multiplication and in standard form. a\) ( 2)^5^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ b\) 10^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_ \_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_ \_\_ c\) Six squared **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** d\) Five cubed **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_** 4\. Evaluate. 2.2 a\) 10^0^ **\_\_\_\_\_\_** b) ( 4)^0^ **\_\_\_\_\_\_** c\) 8^1^ **\_\_\_\_\_\_** d) 4^0^ **\_\_\_\_\_\_** 5\. Write each number in standard form. a\) 9 10^3^ 9 **\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_** 9 **\_\_\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** 90 b\) (1 10^2^) (3 10^1^) (5 10^0^) (1 ) (3 ) (5 ) **\_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_** \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c\) (2 10^3^) (4 10^2^) (1 10^1^) (9 10^0^) (2 **\_\_\_\_\_\_**) (4 **\_\_\_\_\_\_**) (1 **\_\_\_\_\_\_**) (9 **\_\_\_\_\_\_**) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ d\) (5 10^4^) (3 10^2^) (7 10^1^) (2 10^0^) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 6\. Evaluate. 2.3 a\) 3^2^ 3 b) \[( 2) 4)\]^3^ **\_\_\_\_\_\_** 3 **\_\_\_\_\_\_**^3^ **\_\_\_\_\_\_** 3 \_\_\_\_\_\_\_**\_\_\_\_** **\_\_\_\_\_\_** **\_\_\_\_\_\_** c\) (20 5) 5^2^ **\_\_\_\_\_\_** 5^2^ d) (8^2^ 4) (6^2^ 6) **\_\_\_\_\_\_** **\_\_\_\_\_\_** (**\_\_\_\_\_\_** 4) (**\_\_\_\_\_\_** 6) **\_\_\_\_\_\_** **\_\_\_\_\_\_** **\_\_\_\_\_\_** **\_\_\_\_\_\_** 7\. Evaluate. a\) 5 3^2^ 5 **\_\_\_\_\_\_** b) 10 (3^2^ 5^0^) 10 **\_\_\_\_\_\_\_\_\_** **\_\_\_\_\_\_** 10 **\_\_\_\_\_\_** **\_\_\_\_\_\_** c\) ( 2)^3^ ( 3)(4) **\_\_\_\_\_\_\_** **\_\_\_\_\_\_\_** d) ( 3) 4^0^ ( 3) ( 3) **\_\_\_\_\_\_** ( 3) **\_\_\_\_\_\_\_** ( 3) **\_\_\_\_\_\_** **\_\_\_\_ \_ \_\_** 91 8\. Write as a power. 2.4 a\) 6^3^ 6^7^ 6^(\_\_\_\_\ \_\_\_\_)^ b) (--4)^2^ ( 4)^3^ ( 4) ^\_\_\_\_\_\_\_\_^ ~6~^\_\_\_\_^ ~(\ 4)~\_\_\_\_ c\) ( 2)^5^ ( 2)^4^ ( 2) ^\_\_\_\_\_\_\_\_^ d) 10^7^ 10 \_\_\_\_\_\_\_\_\_\_\_\_\_ ~(\ 2)~^\_\_\_\_^ \_\_\_\_\_\_\_\_\_\_\_\_\_ 9\. Write as a power. ^5^ a\) 5^7^ 5^3^ 5^(\_\_\_\_\ \_\_\_\_)^ b) \_\_\_\_\_\_\_\_\_\_\_\_\_ 10^3^ ~5~^\_\_\_\_^ \_\_\_\_\_\_\_\_\_\_\_\_\_ ^10^ c\) ( 6)^8^ ( 6)^2^ \_\_\_\_\_\_\_\_\_\_\_\_\_ d) \_\_\_\_\_\_\_\_\_\_\_\_\_ 5^6^ \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_ [( 3)]^4^ e\) 8^3^ 8 \_\_\_\_\_\_\_\_\_\_\_\_\_ f) \_\_\_\_\_\_\_\_\_\_\_\_\_ ( 3)^0^ \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_ 10\. Write as a power. 2.5 a\) (5^3^)^4^ ~5~\_\_\_\_ ^\_\_\_\_^ b) \[( 3)^2^\]^6^ ~(\ 3)~\_\_\_\_ \_\_\_\_ ~5~^\_\_\_\_^ ~(\ 3)~\_\_\_\_ c\) (8^2^)^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_ d) \[( 5)^5^\]^4^ \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_ 11\. Write as a product or quotient of powers. a\) (3 5)^2^ ~3~^\_\_\_\_^ ~5~^\_\_\_\_^ b) (2 10)^5^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ ~a~^^~3~~~b~~^5^ c\) \[( 4) ( 5)\]^3\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\ d)^ \_\_\_\_\_\_ e\) (12 10)^4^ ~12~^\_\_\_\_^ ~10~^\_\_\_\_^ f) \[( 7) ( 9)\]^6^ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 92